Abstract. Sledgehammer is a highly successful subsystem of Isabelle/HOL that calls automatic theorem provers to assist with interactive proof construction. It requires no user configuration: it can be invoked with a single mouse gesture at any point in a proof. It automatically finds relevant lemmas from all those currently available. An unusual aspect of its architecture is its use of unsound translations, coupled with its delivery of results as Isabelle/HOL proof scripts: its output cannot be trusted, but it does not need to be trusted. Sledgehammer works well with Isar structured proofs and allows beginners to prove challenging theorems. |
Abstract. This note reports on some experiments, using a handful of standard automated reasoning tools, for exploring Steinitz-Rademacher polyhedra, which are models of a certain first-order theory of incidence structures. This theory and its models, even simple ones, presents significant, geometrically fascinating challenges for automated reasoning tools are. |
Abstract. In this paper we use the Kripke semantics characterization of Dummett logic to introduce a new way of handling non-forced formulas in tableau proof systems. We pursue the aim of reducing the search space by strictly increasing the number of forced propositional variables after the application of non-invertible rules. The focus of the paper is on a new tableau system for Dummett logic, for which we have an implementation. The ideas presented can be extended to intuitionistic logic and intermediate logics as well. |
Abstract. Studies on type theory have brought numerous important contributions to computer science. In this paper we present a GUI-based proof tool that provides assistance in constructing deductions in type theory and validating implicational intuitionistic logic formulas. As such, this proof tool is a testbed for learning type theory and implicational intuitionistic logic. This proof tool focuses on an important variant of type theory named TA<sub>λ</sub>, especially on its two core algorithms: the principal-type algorithm and the type inhabitant search algorithm. The former algorithm finds a most general type assignable to a given λ-term, while the latter finds inhabitants (closed λ-terms in β-normal form) to which a given type can be assigned. By the Curry-Howard correspondence, the latter algorithm provides provability for implicational formulas in intuitionistic logic. We elaborate on how to implement those two algorithms declaratively in Prolog and the overall GUI-based program architecture. In our implementation, we make some modification to improve the performance of the principal-type algorithm. We have also built a web-based version of the proof tool called λ-Guru. |
Abstract. We recall the recent approach by (Zankl and Korp, 2010) to prove upper bounds on the (derivational) complexity of term rewrite systems modularly. In this note we show that this approach is suitable to tighten bounds after they have been established. The idea is to replace proof steps with a large bound by (new) proofs that yield smaller bounds. An evaluation of the approach shows the benefits. |
Abstract. This paper presents an optimized algorithm for solving the satisfiability problem (PSAT) in the probabilistic description logic P-SROIQ. P-SROIQ and related Nilsson-style probabilistic logics the PSAT problem is typically solved by reduction to linear programming. This straightforward approach does not scale well because the number of variables in linear programs grows exponentially with the number of probabilistic statements. In this paper we demonstrate an algorithm to cope with this problem which is based on column generation. Although column generation approaches to PSAT have been known for the last two decades, this is, to the best of our knowledge, the first algorithm which also works for a non-propositional probabilistic logic. We report results of an empirical investigation which show that the algorithm can handle probabilistic knowledge bases of about 1000 probabilistic statements in addition to even larger number of classical SROIQ axioms. |
Abstract. Non-trivial linear straight-line programs over the Galois field of two elements occur frequently in applications such as encryption or high-performance computing. Finding the shortest linear straight-line program for a given set of linear forms is known to be MaxSNP-complete, i.e., there is no ε-approximation for the problem unless <math>P = NP</math>.
This paper reiterates a non-approximative approach for finding the shortest linear straight-line program. After showing how to search for a circuit of XOR gates with the minimal number of such gates by a reduction of the associated decision problem ("Is there a program of length <math>k</math>?") to satisfiability of propositional logic, we show that using modern SAT solvers, provably optimal solutions to interesting problem instances from cryptography can be obtained. We substantiate this claim by a case study on optimizing the AES S-Box. |